Nagata's conjecture on a polynomial automorphism in positive characteristic

Abstract

An automorphism of the polynomial ring $k[x_1,\dots ,x_n]$ over a field $k$ is said to be $tame$ if it can be obtained by composing affine automorphisms and elementary automorphisms, and $wild$ otherwise. Jung and van der Kulk showed that every automorphism of $k[x_1,x_2]$ is tame. In 1972, Nagata conjectured that a certain automorphism of $k[x_1,x_2,x_3]$ is wild. In 2003, Shestakov and Umirbaev proved this conjecture for $\mathrm{char}k=0$. The purpose of this paper is to prove the conjecture for $\mathrm{char}k\ge 7$. This is the first time that the existence of a wild automorphism has been confirmed in positive characteristic.

Minor corrections